where the subscript `0' denotes vacuum and \(C\)  is an integration constant. In the last step we substituted \(v_{eff}\)  given by (REF ) and this yields \(\Lambda _{eff}\)  given by (REF ) as the coefficient of \(-r^2\)  (see also Eq. (REF )). Of course, this is exactly what we expect the metric of pure AdS\(_3\)  to be for cosmological constant \(\Lambda _{eff}\) . The initial conditions at \(r=0\)  are determined by the constant \(C\) . We set \(C=1\)  since in \(2+1\)  dimensions this choice avoids a conical singularity at the origin [1]}, [2]}. Moreover, \(C=1\)  also works for the case of vortices embedded in asymptotically Minkowski spacetime (\(\Lambda _{eff}=0\) ).
